Proof Theory in The Light of Categories Giovanni Cin a & - - PowerPoint PPT Presentation

Proof Theory in The Light of Categories

Giovanni Cin´ a & Giuseppe Greco April 5, 2013

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Part 1 - Proof Theory

1

From global- to local-rules calculi Axiomatic Calculi Natural Deduction Calculi Sequent Calculi Cut-elimination

2

From holistic to modular calculi Display Calculi Propositions- and Structures-Language Display Postulates and Display Property Structural Rules Operational Rules No-standard Rules

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From global- to local-rules calculi Axiomatic Calculi

Axiomatic calculi ´ a la Hilbert were the first to appear and, typically, are characterized by ‘more’ axioms and ‘few’ inference rules, at the limit only one (Modus Ponens). The objects manipulated in such calculi are formulas. The meaning of logical symbols is implicitly defined by the axioms that, also, set their mutual relations. Again, the axioms allow only an indirect control of the ‘structure’.

1 (A → ((A → A) → A)) → ((A → (A → A)) → (A → A)) 2 A → ((A → A) → A) 3 (A → (A → A)) → (A → A) 4 A → (A → A) 5 A → A 1 2

MP

3 4

MP

5

where the leaves are all instantiations of axioms.

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From global- to local-rules calculi Axiomatic Calculi

Axiomatic calculi ´ a la Hilbert were the first to appear and, typically, are characterized by ‘more’ axioms and ‘few’ inference rules, at the limit only one (Modus Ponens). The objects manipulated in such calculi are formulas. The meaning of logical symbols is implicitly defined by the axioms that, also, set their mutual relations. Again, the axioms allow only an indirect control of the ‘structure’.

1 (A → ((A → A) → A)) → ((A → (A → A)) → (A → A)) 2 A → ((A → A) → A) 3 (A → (A → A)) → (A → A) 4 A → (A → A) 5 A → A 1 2

MP

3 4

MP

5

where the leaves are all instantiations of axioms.

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From global- to local-rules calculi Axiomatic Calculi

Axiomatic calculi ´ a la Hilbert were the first to appear and, typically, are characterized by ‘more’ axioms and ‘few’ inference rules, at the limit only one (Modus Ponens). The objects manipulated in such calculi are formulas. The meaning of logical symbols is implicitly defined by the axioms that, also, set their mutual relations. Again, the axioms allow only an indirect control of the ‘structure’.

1 (A → ((A → A) → A)) → ((A → (A → A)) → (A → A)) 2 A → ((A → A) → A) 3 (A → (A → A)) → (A → A) 4 A → (A → A) 5 A → A 1 2

MP

3 4

MP

5

where the leaves are all instantiations of axioms.

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From global- to local-rules calculi Axiomatic Calculi

Advantages: proofs on the system are simplified for systems with few and simple inference rules; the space of logics can be reconstructed in a modular way: adding axioms to a previous axiomatization we get other logics. Disadvantages: the proofs in the system are long and often unnatural; the meaning of connectives is global: e.g. the axiom (A → B) → ((C → B) → (A ∨ C → B)) involves different connectives; the derivations are global: e.g. only Modus Ponens is used to prove all theorems.

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From global- to local-rules calculi Axiomatic Calculi

Advantages: proofs on the system are simplified for systems with few and simple inference rules; the space of logics can be reconstructed in a modular way: adding axioms to a previous axiomatization we get other logics. Disadvantages: the proofs in the system are long and often unnatural; the meaning of connectives is global: e.g. the axiom (A → B) → ((C → B) → (A ∨ C → B)) involves different connectives; the derivations are global: e.g. only Modus Ponens is used to prove all theorems.

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From global- to local-rules calculi Natural Deduction Calculi

Natural deduction calculi ´ a la Gentzen are characterized by the use

• f assumptions (introduced by an explicit rule) and different inference

rules for different connectives. The objects manipulated in such calculi are formulas. The meaning of the logical symbols is explicitly defined (by Intr/Elim Rule): an operational content corresponds to each connective. Introduction Rules for implication and negation discharge assumptions: appropriate restrictions allow some control of the ‘structure’.

[A ∧ B]1 [¬A ∨ ¬B]2 [A ∧ B]3

E∧

A [¬A]4

I∧

A ∧ ¬A

3 I¬

¬(A ∧ B) [A ∧ B]5

E∧

B [¬B]6

I∧

B ∧ ¬B

5 I¬

¬(A ∧ B)

4,6 E∨

¬(A ∧ B)

I∧

(A ∧ B) ∧ ¬(A ∧ B)

2 I¬

¬(¬A ∨ ¬B)

1,3,5 I→

A ∧ B → ¬(¬A ∨ ¬B)

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From global- to local-rules calculi Natural Deduction Calculi

Natural deduction calculi ´ a la Gentzen are characterized by the use

• f assumptions (introduced by an explicit rule) and different inference

rules for different connectives. The objects manipulated in such calculi are formulas. The meaning of the logical symbols is explicitly defined (by Intr/Elim Rule): an operational content corresponds to each connective. Introduction Rules for implication and negation discharge assumptions: appropriate restrictions allow some control of the ‘structure’.

[A ∧ B]1 [¬A ∨ ¬B]2 [A ∧ B]3

E∧

A [¬A]4

I∧

A ∧ ¬A

3 I¬

¬(A ∧ B) [A ∧ B]5

E∧

B [¬B]6

I∧

B ∧ ¬B

5 I¬

¬(A ∧ B)

4,6 E∨

¬(A ∧ B)

I∧

(A ∧ B) ∧ ¬(A ∧ B)

2 I¬

¬(¬A ∨ ¬B)

1,3,5 I→

A ∧ B → ¬(¬A ∨ ¬B)

5 / 37

From global- to local-rules calculi Natural Deduction Calculi

Natural deduction calculi ´ a la Gentzen are characterized by the use

• f assumptions (introduced by an explicit rule) and different inference

rules for different connectives. The objects manipulated in such calculi are formulas. The meaning of the logical symbols is explicitly defined (by Intr/Elim Rule): an operational content corresponds to each connective. Introduction Rules for implication and negation discharge assumptions: appropriate restrictions allow some control of the ‘structure’.

[A ∧ B]1 [¬A ∨ ¬B]2 [A ∧ B]3

E∧

A [¬A]4

I∧

A ∧ ¬A

3 I¬

¬(A ∧ B) [A ∧ B]5

E∧

B [¬B]6

I∧

B ∧ ¬B

5 I¬

¬(A ∧ B)

4,6 E∨

¬(A ∧ B)

I∧

(A ∧ B) ∧ ¬(A ∧ B)

2 I¬

¬(¬A ∨ ¬B)

1,3,5 I→

A ∧ B → ¬(¬A ∨ ¬B)

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From global- to local-rules calculi Natural Deduction Calculi

Advantages: the proofs in the system are natural; the connectives are introduced one by one (this is in the direction

• f proof-theoretic semantics);

Disadvantages: assumptions tipically are discharged after many steps in a derivation; it is not simple to reconstruct the space of the logics; it is difficult to obtain natural deduction calculi for non-classical or modal logics.

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From global- to local-rules calculi Natural Deduction Calculi

Advantages: the proofs in the system are natural; the connectives are introduced one by one (this is in the direction

• f proof-theoretic semantics);

Disadvantages: assumptions tipically are discharged after many steps in a derivation; it is not simple to reconstruct the space of the logics; it is difficult to obtain natural deduction calculi for non-classical or modal logics.

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From global- to local-rules calculi Sequent Calculi

Sequent calculi ´ a la Gentzen are characterized by a single axiom (Identity), the use of assumptions and conclusions, by different inference rules for different connectives and for different structural

• perations.

Objects manipulated in such calculations are sequents: Γ ⊢ ∆ where Γ and ∆ are (possibly empty) sequences of formulas separated by a (poliadyc) comma. The meaning of logical symbols is explicitly defined (by Left/Right Introduction Rule). The structural rules allow a direct control of the ‘structure’.

A ⊢ A ⊥ ⊢ ⊥

W

A, ⊥ ⊢ ⊥ A, A → ⊥ ⊢ ⊥ A, ¬A ⊢ ⊥ A ∧ B, ¬A ⊢ ⊥ B ⊢ B ⊥ ⊢ ⊥

W

B, ⊥ ⊢ ⊥ B, B → ⊥ ⊢ ⊥ B, ¬B ⊢ ⊥ A ∧ B, ¬B ⊢ ⊥ A ∧ B, ¬A ∨ ¬B ⊢ ⊥

E

¬A ∨ ¬B, A ∧ B ⊢ ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

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From global- to local-rules calculi Sequent Calculi

Sequent calculi ´ a la Gentzen are characterized by a single axiom (Identity), the use of assumptions and conclusions, by different inference rules for different connectives and for different structural

• perations.

Objects manipulated in such calculations are sequents: Γ ⊢ ∆ where Γ and ∆ are (possibly empty) sequences of formulas separated by a (poliadyc) comma. The meaning of logical symbols is explicitly defined (by Left/Right Introduction Rule). The structural rules allow a direct control of the ‘structure’.

A ⊢ A ⊥ ⊢ ⊥

W

A, ⊥ ⊢ ⊥ A, A → ⊥ ⊢ ⊥ A, ¬A ⊢ ⊥ A ∧ B, ¬A ⊢ ⊥ B ⊢ B ⊥ ⊢ ⊥

W

B, ⊥ ⊢ ⊥ B, B → ⊥ ⊢ ⊥ B, ¬B ⊢ ⊥ A ∧ B, ¬B ⊢ ⊥ A ∧ B, ¬A ∨ ¬B ⊢ ⊥

E

¬A ∨ ¬B, A ∧ B ⊢ ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

7 / 37

From global- to local-rules calculi Sequent Calculi

Sequent calculi ´ a la Gentzen are characterized by a single axiom (Identity), the use of assumptions and conclusions, by different inference rules for different connectives and for different structural

• perations.

Objects manipulated in such calculations are sequents: Γ ⊢ ∆ where Γ and ∆ are (possibly empty) sequences of formulas separated by a (poliadyc) comma. The meaning of logical symbols is explicitly defined (by Left/Right Introduction Rule). The structural rules allow a direct control of the ‘structure’.

A ⊢ A ⊥ ⊢ ⊥

W

A, ⊥ ⊢ ⊥ A, A → ⊥ ⊢ ⊥ A, ¬A ⊢ ⊥ A ∧ B, ¬A ⊢ ⊥ B ⊢ B ⊥ ⊢ ⊥

W

B, ⊥ ⊢ ⊥ B, B → ⊥ ⊢ ⊥ B, ¬B ⊢ ⊥ A ∧ B, ¬B ⊢ ⊥ A ∧ B, ¬A ∨ ¬B ⊢ ⊥

E

¬A ∨ ¬B, A ∧ B ⊢ ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

7 / 37

From global- to local-rules calculi Sequent Calculi

Advantages: the derivations are local; the proofs in the system are automatizable (if the calculus enjoy cut-elimination); a distinction between connectives and structure is introduced (this is in the direction of proof-theoretic semantics). Disadvantages: the space of logics cannot be reconstructed in a modular way (if the calculus is non-standard, i.e. as usual for modal logics); it is not simple to obtain sequent calculi for substructural or modal logics (with the sub-formula property).

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From global- to local-rules calculi Sequent Calculi

Advantages: the derivations are local; the proofs in the system are automatizable (if the calculus enjoy cut-elimination); a distinction between connectives and structure is introduced (this is in the direction of proof-theoretic semantics). Disadvantages: the space of logics cannot be reconstructed in a modular way (if the calculus is non-standard, i.e. as usual for modal logics); it is not simple to obtain sequent calculi for substructural or modal logics (with the sub-formula property).

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From global- to local-rules calculi Cut-elimination

Common forms of the cut rule are the following:

Γ ⊢ C, ∆ Γ′, C ⊢ ∆′ Γ′, Γ ⊢ ∆′, ∆ Γ ⊢ C, ∆ Γ, C ⊢ ∆ Γ ⊢ ∆ Γ ⊢ C Γ′, C ⊢ ∆ Γ′, Γ ⊢ ∆ Γ ⊢ C, ∆ C ⊢ ∆′ Γ ⊢ ∆′, ∆

Theorem (Cut-elimination) If Γ ⊢ ∆ is derivable in the calculus S with Cut, then it is in S without Cut. The cut-elimination is the most fundamental technique in proof theory and many important syntactic properties derive from it (e.g. decidability). A cut is an intermediate step in a deduction, by which a conclusion(s) ∆ can be proved from the assumption(s) Γ via the lemma C. ‘Eliminating the cut’ from such a proof generates a new (and lemma-free) proof of ∆, which exclusively employs syntactic material coming from Γ and ∆ (subformula property). Typically, syntactic proofs of cut-elimination are non-modular: if a new rule is added, cut-elimination must be proved from scratch.

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From global- to local-rules calculi Cut-elimination

Common forms of the cut rule are the following:

Γ ⊢ C, ∆ Γ′, C ⊢ ∆′ Γ′, Γ ⊢ ∆′, ∆ Γ ⊢ C, ∆ Γ, C ⊢ ∆ Γ ⊢ ∆ Γ ⊢ C Γ′, C ⊢ ∆ Γ′, Γ ⊢ ∆ Γ ⊢ C, ∆ C ⊢ ∆′ Γ ⊢ ∆′, ∆

Theorem (Cut-elimination) If Γ ⊢ ∆ is derivable in the calculus S with Cut, then it is in S without Cut. The cut-elimination is the most fundamental technique in proof theory and many important syntactic properties derive from it (e.g. decidability). A cut is an intermediate step in a deduction, by which a conclusion(s) ∆ can be proved from the assumption(s) Γ via the lemma C. ‘Eliminating the cut’ from such a proof generates a new (and lemma-free) proof of ∆, which exclusively employs syntactic material coming from Γ and ∆ (subformula property). Typically, syntactic proofs of cut-elimination are non-modular: if a new rule is added, cut-elimination must be proved from scratch.

9 / 37

From global- to local-rules calculi Cut-elimination

Common forms of the cut rule are the following:

Γ ⊢ C, ∆ Γ′, C ⊢ ∆′ Γ′, Γ ⊢ ∆′, ∆ Γ ⊢ C, ∆ Γ, C ⊢ ∆ Γ ⊢ ∆ Γ ⊢ C Γ′, C ⊢ ∆ Γ′, Γ ⊢ ∆ Γ ⊢ C, ∆ C ⊢ ∆′ Γ ⊢ ∆′, ∆

Theorem (Cut-elimination) If Γ ⊢ ∆ is derivable in the calculus S with Cut, then it is in S without Cut. The cut-elimination is the most fundamental technique in proof theory and many important syntactic properties derive from it (e.g. decidability). A cut is an intermediate step in a deduction, by which a conclusion(s) ∆ can be proved from the assumption(s) Γ via the lemma C. ‘Eliminating the cut’ from such a proof generates a new (and lemma-free) proof of ∆, which exclusively employs syntactic material coming from Γ and ∆ (subformula property). Typically, syntactic proofs of cut-elimination are non-modular: if a new rule is added, cut-elimination must be proved from scratch.

9 / 37

From global- to local-rules calculi Cut-elimination

Common forms of the cut rule are the following:

Γ ⊢ C, ∆ Γ′, C ⊢ ∆′ Γ′, Γ ⊢ ∆′, ∆ Γ ⊢ C, ∆ Γ, C ⊢ ∆ Γ ⊢ ∆ Γ ⊢ C Γ′, C ⊢ ∆ Γ′, Γ ⊢ ∆ Γ ⊢ C, ∆ C ⊢ ∆′ Γ ⊢ ∆′, ∆

Theorem (Cut-elimination) If Γ ⊢ ∆ is derivable in the calculus S with Cut, then it is in S without Cut. The cut-elimination is the most fundamental technique in proof theory and many important syntactic properties derive from it (e.g. decidability). A cut is an intermediate step in a deduction, by which a conclusion(s) ∆ can be proved from the assumption(s) Γ via the lemma C. ‘Eliminating the cut’ from such a proof generates a new (and lemma-free) proof of ∆, which exclusively employs syntactic material coming from Γ and ∆ (subformula property). Typically, syntactic proofs of cut-elimination are non-modular: if a new rule is added, cut-elimination must be proved from scratch.

9 / 37

From holistic to modular calculi Display Calculi

Display calculi were introduced by Belnap [1.2] to provide a uniform account for cut-elimination; a ‘pure’ proof-theoretical analysis of logics; a tool useful to ‘merge’ different logics. Display calculi generalize sequent calculi allowing: different ‘structural connectives’ (not just the Gentzen’s comma), where the structures in X ⊢ Y are binary trees (not sequences); a set of structural rules named Display Postulates, that give the Display Property (essential in Belnap’s cut-elimination).

A ⊢ A A ; B ⊢ A A ∧ B ⊢ A ⊥ ⊢ ⊥ A → ⊥ ⊢ A ∧ B > ⊥ ¬A ⊢ A ∧ B > ⊥ B ⊢ B A ; B ⊢ B A ∧ B ⊢ B ⊥ ⊢ ⊥ B → ⊥ ⊢ A ∧ B > ⊥ ¬B ⊢ A ∧ B > ⊥ ¬A ∨ ¬B ⊢ (A ∧ B > ⊥) ; (A ∧ B > ⊥) ¬A ∨ ¬B ⊢ A ∧ B > ⊥

> ;

A ∧ B ; ¬A ∨ ¬B ⊢ ⊥ ¬A ∨ ¬B ; A ∧ B ⊢ ⊥

; >

A ∧ B ⊢ ¬A ∨ ¬B > ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

10 / 37

From holistic to modular calculi Display Calculi

Display calculi were introduced by Belnap [1.2] to provide a uniform account for cut-elimination; a ‘pure’ proof-theoretical analysis of logics; a tool useful to ‘merge’ different logics. Display calculi generalize sequent calculi allowing: different ‘structural connectives’ (not just the Gentzen’s comma), where the structures in X ⊢ Y are binary trees (not sequences); a set of structural rules named Display Postulates, that give the Display Property (essential in Belnap’s cut-elimination).

A ⊢ A A ; B ⊢ A A ∧ B ⊢ A ⊥ ⊢ ⊥ A → ⊥ ⊢ A ∧ B > ⊥ ¬A ⊢ A ∧ B > ⊥ B ⊢ B A ; B ⊢ B A ∧ B ⊢ B ⊥ ⊢ ⊥ B → ⊥ ⊢ A ∧ B > ⊥ ¬B ⊢ A ∧ B > ⊥ ¬A ∨ ¬B ⊢ (A ∧ B > ⊥) ; (A ∧ B > ⊥) ¬A ∨ ¬B ⊢ A ∧ B > ⊥

> ;

A ∧ B ; ¬A ∨ ¬B ⊢ ⊥ ¬A ∨ ¬B ; A ∧ B ⊢ ⊥

; >

A ∧ B ⊢ ¬A ∨ ¬B > ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

10 / 37

From holistic to modular calculi Display Calculi

Display calculi were introduced by Belnap [1.2] to provide a uniform account for cut-elimination; a ‘pure’ proof-theoretical analysis of logics; a tool useful to ‘merge’ different logics. Display calculi generalize sequent calculi allowing: different ‘structural connectives’ (not just the Gentzen’s comma), where the structures in X ⊢ Y are binary trees (not sequences); a set of structural rules named Display Postulates, that give the Display Property (essential in Belnap’s cut-elimination).

A ⊢ A A ; B ⊢ A A ∧ B ⊢ A ⊥ ⊢ ⊥ A → ⊥ ⊢ A ∧ B > ⊥ ¬A ⊢ A ∧ B > ⊥ B ⊢ B A ; B ⊢ B A ∧ B ⊢ B ⊥ ⊢ ⊥ B → ⊥ ⊢ A ∧ B > ⊥ ¬B ⊢ A ∧ B > ⊥ ¬A ∨ ¬B ⊢ (A ∧ B > ⊥) ; (A ∧ B > ⊥) ¬A ∨ ¬B ⊢ A ∧ B > ⊥

> ;

A ∧ B ; ¬A ∨ ¬B ⊢ ⊥ ¬A ∨ ¬B ; A ∧ B ⊢ ⊥

; >

A ∧ B ⊢ ¬A ∨ ¬B > ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

10 / 37

From holistic to modular calculi Display Calculi

Advantages: cut-elimination is a consequence of design principles, by the following: Theorem (Cut-elimination [1.2] [1.5]) If a logic is ‘properly displayable’, then it enjoys cut-elimination space of logics can be reconstructed in a modular way, because

• f:

Doˇ sen Principle [1.5] The rules for the logical operations are never changed: all changes are made in the structural rules a ‘real’ proof-theory is possible for substrucural and modal logics (e.g. separated, symmetrical and explicit introduction rules for the normal modal operators are available). Disadvantages: not amenable for proof-search (because of Display Postulates).

11 / 37

From holistic to modular calculi Display Calculi

Advantages: cut-elimination is a consequence of design principles, by the following: Theorem (Cut-elimination [1.2] [1.5]) If a logic is ‘properly displayable’, then it enjoys cut-elimination space of logics can be reconstructed in a modular way, because

• f:

Doˇ sen Principle [1.5] The rules for the logical operations are never changed: all changes are made in the structural rules a ‘real’ proof-theory is possible for substrucural and modal logics (e.g. separated, symmetrical and explicit introduction rules for the normal modal operators are available). Disadvantages: not amenable for proof-search (because of Display Postulates).

11 / 37

From holistic to modular calculi Propositions- and Structures-Language

As case study, we consider the display calculus (plus explicit negations) introduced in Greco, Kurz, Palmigiano [1.3] for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge EAK [1.1]. For each agent a ∈ Ag and action α ∈ Act, Propositions are built from a set of atomic propositional variables AtProp = {p, q, r, . . .} and two constants ⊥ and ⊤: A := p | ⊥ | ⊤ | A ∧ A | A ∨ A | A → A | A > A | ¬A | ∼A | ✸

aA | ✷aA | aA | aA | [α]A | αA |

• α
• A |

[ α ] A . Structures are built from formulas and one structural constant I: X :=

• I | A | X; X | X > X | ∗X |
• aX | ◦

aX | {α}X |

{ α } X .

12 / 37

From holistic to modular calculi Propositions- and Structures-Language

As case study, we consider the display calculus (plus explicit negations) introduced in Greco, Kurz, Palmigiano [1.3] for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge EAK [1.1]. For each agent a ∈ Ag and action α ∈ Act, Propositions are built from a set of atomic propositional variables AtProp = {p, q, r, . . .} and two constants ⊥ and ⊤: A := p | ⊥ | ⊤ | A ∧ A | A ∨ A | A → A | A > A | ¬A | ∼A | ✸

aA | ✷aA | aA | aA | [α]A | αA |

• α
• A |

[ α ] A . Structures are built from formulas and one structural constant I: X :=

• I | A | X; X | X > X | ∗X |
• aX | ◦

aX | {α}X |

{ α } X .

12 / 37

From holistic to modular calculi Propositions- and Structures-Language

As case study, we consider the display calculus (plus explicit negations) introduced in Greco, Kurz, Palmigiano [1.3] for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge EAK [1.1]. For each agent a ∈ Ag and action α ∈ Act, Propositions are built from a set of atomic propositional variables AtProp = {p, q, r, . . .} and two constants ⊥ and ⊤: A := p | ⊥ | ⊤ | A ∧ A | A ∨ A | A → A | A > A | ¬A | ∼A | ✸

aA | ✷aA | aA | aA | [α]A | αA |

• α
• A |

[ α ] A . Structures are built from formulas and one structural constant I: X :=

• I | A | X; X | X > X | ∗X |
• aX | ◦

aX | {α}X |

{ α } X .

12 / 37

From holistic to modular calculi Propositions- and Structures-Language

The structural connectives are contextual (as the Gentzen’s comma) and each of them is associated with a pair of logical connectives: Structural symb: I ; > ∗ Operational symb: ⊤ ⊥ ∧ ∨

>

→ ¬ ∼ Structural symb:

• a
• a

{α} { α } Operational symb: ✸

a

✷a

• a

a α [α]

• α
• [

α ] by the translations τ1 of precedent and τ2 of succedent into prop. :

τ1(A) := A τ2(A) := A τ1(I) := ⊤ τ2(I) := ⊥ τ1(X ; Y) := τ1(X) ∧ τ1(Y) τ2(X ; Y) := τ2(X) ∨ τ2(Y) τ1(X > Y) := τ2(X) > τ1(Y) τ2(X > Y) := τ1(X) → τ2(Y) τ1(∗X) := ∼τ2(X) τ2(∗X) := ¬τ1(X) τ1(◦

aX)

:= ✸

aτ1(X)

τ2(◦

aX)

:= ✷aτ2(X) τ1(•

aX)

:=

• aτ1(X)

τ2(•

aX)

:= aτ2(X) τ1({α}X) := ατ1(X) τ2({α}X) := ατ2(X) τ1( { α } X) :=

• α
• τ1(X)

τ2({α}X) := [ α ] τ2(X)

13 / 37

From holistic to modular calculi Propositions- and Structures-Language

The structural connectives are contextual (as the Gentzen’s comma) and each of them is associated with a pair of logical connectives: Structural symb: I ; > ∗ Operational symb: ⊤ ⊥ ∧ ∨

>

→ ¬ ∼ Structural symb:

• a
• a

{α} { α } Operational symb: ✸

a

✷a

• a

a α [α]

• α
• [

α ] by the translations τ1 of precedent and τ2 of succedent into prop. :

τ1(A) := A τ2(A) := A τ1(I) := ⊤ τ2(I) := ⊥ τ1(X ; Y) := τ1(X) ∧ τ1(Y) τ2(X ; Y) := τ2(X) ∨ τ2(Y) τ1(X > Y) := τ2(X) > τ1(Y) τ2(X > Y) := τ1(X) → τ2(Y) τ1(∗X) := ∼τ2(X) τ2(∗X) := ¬τ1(X) τ1(◦

aX)

:= ✸

aτ1(X)

τ2(◦

aX)

:= ✷aτ2(X) τ1(•

aX)

:=

• aτ1(X)

τ2(•

aX)

:= aτ2(X) τ1({α}X) := ατ1(X) τ2({α}X) := ατ2(X) τ1( { α } X) :=

• α
• τ1(X)

τ2({α}X) := [ α ] τ2(X)

13 / 37

From holistic to modular calculi Display Postulates and Display Property

Display Postulates X ; Y ⊢ Z

; >

Y ⊢ X > Z Z ⊢ Y ; X

> ;

Y > Z ⊢ X

• aX ⊢ Y
• a
• a

X ⊢ •

aY

X ⊢ ◦

aY

• a
• a
• aX ⊢ Y

{α}X ⊢ Y

{α} { α }

X ⊢ { α } Y X ⊢ {α}Y

{ α } {α}

{ α } X ⊢ Y ∗X ⊢ Y

∗ ∗L ∗Y ⊢ X

Y ⊢ ∗X

∗ ∗R

X ⊢ ∗Y Z ⊢ Y ; X

; ∗ ;

∗Y ; Z ⊢ X X ; Y ⊢ Z

; ; ∗

Y ⊢ ∗X ; Z ∗∗X ⊢ Y

∗∗L

X ⊢ Y Y ⊢ ∗∗X

∗∗R

Y ⊢ X

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From holistic to modular calculi Display Postulates and Display Property

Display Postulates X ; Y ⊢ Z

; >

Y ⊢ X > Z Z ⊢ Y ; X

> ;

Y > Z ⊢ X

• aX ⊢ Y
• a
• a

X ⊢ •

aY

X ⊢ ◦

aY

• a
• a
• aX ⊢ Y

{α}X ⊢ Y

{α} { α }

X ⊢ { α } Y X ⊢ {α}Y

{ α } {α}

{ α } X ⊢ Y ∗X ⊢ Y

∗ ∗L ∗Y ⊢ X

Y ⊢ ∗X

∗ ∗R

X ⊢ ∗Y Z ⊢ Y ; X

; ∗ ;

∗Y ; Z ⊢ X X ; Y ⊢ Z

; ; ∗

Y ⊢ ∗X ; Z ∗∗X ⊢ Y

∗∗L

X ⊢ Y Y ⊢ ∗∗X

∗∗R

Y ⊢ X

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From holistic to modular calculi Display Postulates and Display Property

By definition, structural connectives form adjoint pairs as follows: ; ⊣ > > ⊣ ;

• a ⊣ •

a

• a ⊣ ◦

a

∗ ⊣ ∗ (1) Nota Bene: ‘adjointness’ in Part 2. So, Display Postulates are ‘about the connection between left and right side of the turnstile’.

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From holistic to modular calculi Display Postulates and Display Property

By definition, structural connectives form adjoint pairs as follows: ; ⊣ > > ⊣ ;

• a ⊣ •

a

• a ⊣ ◦

a

∗ ⊣ ∗ (1) Nota Bene: ‘adjointness’ in Part 2. So, Display Postulates are ‘about the connection between left and right side of the turnstile’.

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From holistic to modular calculi Display Postulates and Display Property

The Display Postulates allow to disassembly and reassembly structures and provide the following: Theorem (Display Property [1.2] [1.5]) Each substructure in a display-sequent is isolable or ‘displayable’ in precedent or, exclusively, succedent position. Note that ‘in precedent (succedent) position’ and ‘on the left (right) side

• f turnstile’ coincide in a Gentzen’s sequent calculus, but not in a

display calculus. E.g. In ‘Y ⊢ X > Z’, X is on the right of the turnstile but it is in a precedent position, in fact it is displayable on the left side: Y ⊢ X > Z X ; Y ⊢ Z Y ; X ⊢ Z X ⊢ Y > Z

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From holistic to modular calculi Display Postulates and Display Property

The Display Postulates allow to disassembly and reassembly structures and provide the following: Theorem (Display Property [1.2] [1.5]) Each substructure in a display-sequent is isolable or ‘displayable’ in precedent or, exclusively, succedent position. Note that ‘in precedent (succedent) position’ and ‘on the left (right) side

• f turnstile’ coincide in a Gentzen’s sequent calculus, but not in a

display calculus. E.g. In ‘Y ⊢ X > Z’, X is on the right of the turnstile but it is in a precedent position, in fact it is displayable on the left side: Y ⊢ X > Z X ; Y ⊢ Z Y ; X ⊢ Z X ⊢ Y > Z

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From holistic to modular calculi Structural Rules

Let be ⊙ ∈ {◦

a, • a}.

Structural Rules

• entry/exit rules -

Id p ⊢ p

X ⊢ A A ⊢ Y

Cut

X ⊢ Y X ⊢ Y

IL X ; I ⊢ Y

Y ⊢ X

IR

Y ⊢ I ; X X ⊢ Z

WL X ; Y ⊢ Z

Z ⊢ Y

WR

Z ⊢ Y ; X X ; X ⊢ Y

CL

X ⊢ Y Y ⊢ X ; X

CR

Y ⊢ X X ⊢ I

⊙ I

⊙X ⊢ I I ⊢ X

I ⊙

I ⊢ ⊙X I ⊢ X

∗

I

∗X ⊢ I X ⊢ I

I

∗

I ⊢ ∗X

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From holistic to modular calculi Structural Rules

Let be ⊛ ∈ {∗, ◦

a, • a, {α},

{ α } }. ⊛X ; ⊛Y ⊢ Z

⊛ ;

⊛(X ; Y) ⊢ Z Z ⊢ ⊛Y ; ⊛X

; ⊛

Z ⊢ ⊛(Y ; X) ⊛X > ⊛Y ⊢ Z

⊛ > ⊛(X > Y) ⊢ Z

Z ⊢ ⊛Y > ⊛X

> ⊛

Z ⊢ ⊛(Y > X)

• manipulation rules -

Y ; X ⊢ Z

EL X ; Y ⊢ Z

Z ⊢ X ; Y

ER

Z ⊢ Y ; X X ; (Y ; Z) ⊢ W

AL (X ; Y) ; Z ⊢ W

W ⊢ (Z ; Y) ; X

AR

W ⊢ Z ; (Y ; X) X > (Y ; Z) ⊢ W

GrnL (X > Y) ; Z ⊢ W

W ⊢ X > (Y ; Z)

GrnR

W ⊢ (X > Y) ; Z

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From holistic to modular calculi Structural Rules

(2) Nota Bene: ‘naturality’ in Part 2. So, Structural Rules are ‘about the left side or, exclusively, the right side of the turnstile’. Note that the Excluded Middle is derivable by Grishin’s rules as follows:

A ⊢ A A ; I ⊢ A A ; I ⊢ ⊥ ; A I ⊢ A > (⊥ ; A)

Grn

I ⊢ (A > ⊥) ; A I ⊢ A ; (A > ⊥) A > I ⊢ A > ⊥ A > I ⊢ A → ⊥ A > I ⊢ ¬A I ⊢ A ; ¬A I ⊢ A ∨ ¬A

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From holistic to modular calculi Structural Rules

(2) Nota Bene: ‘naturality’ in Part 2. So, Structural Rules are ‘about the left side or, exclusively, the right side of the turnstile’. Note that the Excluded Middle is derivable by Grishin’s rules as follows:

A ⊢ A A ; I ⊢ A A ; I ⊢ ⊥ ; A I ⊢ A > (⊥ ; A)

Grn

I ⊢ (A > ⊥) ; A I ⊢ A ; (A > ⊥) A > I ⊢ A > ⊥ A > I ⊢ A → ⊥ A > I ⊢ ¬A I ⊢ A ; ¬A I ⊢ A ∨ ¬A

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From holistic to modular calculi Operational Rules

Operational Rules

• translation rules -

⊥L ⊥ ⊢ I

X ⊢ I

⊥R

X ⊢ ⊥ I ⊢ X

⊤L ⊤ ⊢ X ⊤R

I ⊢ ⊤ A ; B ⊢ Z

∧L A ∧ B ⊢ Z

X ⊢ A Y ⊢ B

∧R

X ; Y ⊢ A ∧ B B ⊢ Y A ⊢ X

∨L

B ∨ A ⊢ Y ; X Z ⊢ B ; A

∨R

Z ⊢ B ∨ A X ⊢ A B ⊢ Y

→L

A → B ⊢ X > Y Z ⊢ A > B

→R

Z ⊢ A → B A > B ⊢ Z

> L A > B ⊢ Z

Y ⊢ B A ⊢ X

> R

X > Y ⊢ A > B

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From holistic to modular calculi Operational Rules

Let be ⊙α ∈ {◦

a, • a, {α},

{ α } }, ✸ ·α ∈ {✸

a, a, α,

• α
• },

⊡α ∈ {✷a, a, [α], [ α ] }. ⊙αA ⊢ X

✸ ·αL

✸ ·αA ⊢ X X ⊢ A

✸ ·αR

⊙αX ⊢ ✸ ·αA A ⊢ X

⊡αL ⊡αA ⊢ ⊙αX

X ⊢ ⊙αA ⊡αR X ⊢ ⊡αA ∗A ⊢ X

∼L ∼A ⊢ X

A ⊢ X

∼R

∗X ⊢ ∼A X ⊢ A

¬L ¬A ⊢ ∗X

X ⊢ ∗A

¬R

X ⊢ ¬A (3) Nota Bene: ‘functoriality’ in Part 2. So, (one half of the) Operational Rules are ‘about left and right side of the turnstile at the same time’.

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From holistic to modular calculi Operational Rules

Let be ⊙α ∈ {◦

a, • a, {α},

{ α } }, ✸ ·α ∈ {✸

a, a, α,

• α
• },

⊡α ∈ {✷a, a, [α], [ α ] }. ⊙αA ⊢ X

✸ ·αL

✸ ·αA ⊢ X X ⊢ A

✸ ·αR

⊙αX ⊢ ✸ ·αA A ⊢ X

⊡αL ⊡αA ⊢ ⊙αX

X ⊢ ⊙αA ⊡αR X ⊢ ⊡αA ∗A ⊢ X

∼L ∼A ⊢ X

A ⊢ X

∼R

∗X ⊢ ∼A X ⊢ A

¬L ¬A ⊢ ∗X

X ⊢ ∗A

¬R

X ⊢ ¬A (3) Nota Bene: ‘functoriality’ in Part 2. So, (one half of the) Operational Rules are ‘about left and right side of the turnstile at the same time’.

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From holistic to modular calculi No-standard Rules

In a context whit Pre(α), we allow the following no-standard rules. Contextual Operational Rules

• translation rules -

Pre(α) ; {α}A ⊢ X

reverseL

Pre(α) ; [α]A ⊢ X X ⊢ Pre(α) > {α}A reverseR X ⊢ Pre(α) > αA

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From holistic to modular calculi No-standard Rules

Contextual Structural Rules

• entry/exit rules -

X ⊢ Y

balance

{α}X ⊢ {α}Y

atomL {α} p ⊢ p atomR

p ⊢ {α} p Pre(α) ; {α}A ⊢ X

reduceL

{α}A ⊢ X X ⊢ Pre(α) > {α}A

reduceR

X ⊢ {α}A

• manipulation rules -

Pre(α) ; {α}◦

aX ⊢ Y

swap-inL Pre(α) ; ◦

a{β}αaβ X ⊢ Y

Y ⊢ Pre(α) > {α}◦

aX

swap-inR

Y ⊢ Pre(α) > ◦

a{β}αaβ X

• Pre(α) ; ◦

a{β} X ⊢ Y | αaβ

• s-outL

Pre(α) ; {α}◦

aX ⊢ ;

• Y | αaβ
• Y ⊢ Pre(α) > ◦

a{β} X | αaβ

• s-outR

;

• Y | αaβ
• ⊢ Pre(α) > {α}◦

aX

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References Part 1

[1.1] A. Baltag, L.S. Moss, S. Solecki, The logic of public announcements, common knowledge and private suspicions, TARK, 43-56, 1998 [1.2] N. Belnap, Display logic, Journal of Philosophical Logic, 11: 375-417, 1982 [1.3] G. Greco, A. Kurz, A. Palmigiano, Dynamic Epistemic Logic Displayed, Submitted, 2013. [1.4] R. Gor´ e, L. Postniece, A. Tiu, Cut-elimination and Proof Search for Bi-Intuitionistic Tense Logic, Proc. Adv. in Modal Logic, 156-177, 2010 [1.5] H. Wansing, Displaying modal logic, Kluwer Academic Publishers, 1998

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Outline

Part 2 - Category Theory

3

Basic notions

4

Link to Display calculi

5

Framework

6

Example

7

Conclusions Beware: we will be sloppy and intuitive on the technical details. Main reference: S. Awodey. Category Theory, Oxford Logic Guides,

• vol. 49. Oxford: Oxford University Press, 2006.

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Basic notions

Categories and functors

Definition A category C is made of

• bjects A, B, C, . . .

arrows f : A → B, g : A → C, . . . Arrows are closed under composition (when target and source match) and composition of arrows is associative. Every object A has an identity arrow 1A that works as the unit of the composition. Definition A functor F : C → D is a pair of maps (F1, F2) such that F1 maps object of C in objects of D F1 maps arrows of C in arrows of D and also preserves sources and targets, identities and compositions.

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Basic notions

Categories and functors

Definition A category C is made of

• bjects A, B, C, . . .

arrows f : A → B, g : A → C, . . . Arrows are closed under composition (when target and source match) and composition of arrows is associative. Every object A has an identity arrow 1A that works as the unit of the composition. Definition A functor F : C → D is a pair of maps (F1, F2) such that F1 maps object of C in objects of D F1 maps arrows of C in arrows of D and also preserves sources and targets, identities and compositions.

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Basic notions

Definition Given C and D, the product category C × D has as objects pairs of objects (C, D), with C in C and D in D as arrows pairs of arrows (f, f ′), with f in C and f ′ in D In a category B the product of two objects A, B is an object A × B equipped with two arrows π1 : A × B → A and π2 : A × B → B (projections) such that ∀C, f1 : C → A, f2C → B ∃!g : C → A × B that makes the following diagram commute C A A × B B

f1 f2 g π1 π2

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Basic notions

Definition Given C and D, the product category C × D has as objects pairs of objects (C, D), with C in C and D in D as arrows pairs of arrows (f, f ′), with f in C and f ′ in D In a category B the product of two objects A, B is an object A × B equipped with two arrows π1 : A × B → A and π2 : A × B → B (projections) such that ∀C, f1 : C → A, f2C → B ∃!g : C → A × B that makes the following diagram commute C A A × B B

f1 f2 g π1 π2

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Basic notions

Natural transformations

Definition Given two functors F, G : C → D, a natural transformation η : F → G is a family of arrows in D indexed by the objects of C such that, for every arrow f : C → B in C, in D we have F(C) F(B) G(C) G(B)

F(f) ηC ηB G(f)

If all the arrows in the family η are isomorphisms, we call η a natural isomorphism.

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Basic notions

Adjoints

Definition Given two functors F : C → D and G : D → C we say that F is left adjoint of G, in symbols F ⊣ G, if ∀C in C and D in D there is a bijective correspondence between arrows F(C) → D in D and arrows C → G(B) in C. This is usually written F(C) → D C → G(B) Moreover, this bijection is natural both in C and D.

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Link to Display calculi

Why do we need all this? The core idea is the following: display calculi are calculi for arrows in the sense that proofs are seen as arrows, A ⊢1 B C ⊢2 D means there is a unique way to build ⊢2 from ⊢1 and A ⊢1 B C ⊢2 D means there is a unique way to build ⊢2 from ⊢1 and viceversa.

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Link to Display calculi

Why do we need all this? The core idea is the following: display calculi are calculi for arrows in the sense that proofs are seen as arrows, A ⊢1 B C ⊢2 D means there is a unique way to build ⊢2 from ⊢1 and A ⊢1 B C ⊢2 D means there is a unique way to build ⊢2 from ⊢1 and viceversa.

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Link to Display calculi

Why do we need all this? The core idea is the following: display calculi are calculi for arrows in the sense that proofs are seen as arrows, A ⊢1 B C ⊢2 D means there is a unique way to build ⊢2 from ⊢1 and A ⊢1 B C ⊢2 D means there is a unique way to build ⊢2 from ⊢1 and viceversa.

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Link to Display calculi

Why do we need all this? The core idea is the following: display calculi are calculi for arrows in the sense that proofs are seen as arrows, A ⊢1 B C ⊢2 D means there is a unique way to build ⊢2 from ⊢1 and A ⊢1 B C ⊢2 D means there is a unique way to build ⊢2 from ⊢1 and viceversa.

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Link to Display calculi

We have seen that in Display Calculi we have Operational Rules, Structural Rules and Display Rules; each having a particular shape. The shapes are, respectively C ⊢ D

• 1. F(C) ⊢ F(D)

C ⊢ F(D)

• 2. C ⊢ G(D)

F(C) ⊢ D

• 3. C ⊢ G(D)

When we read these as instructions to build arrows, we have that rules of type 1 are given by functoriality rules of type 2 are given by naturality rules of type 3 are given by adjointness (Note: we also have rules of type 2 with functors on the left)

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Link to Display calculi

We have seen that in Display Calculi we have Operational Rules, Structural Rules and Display Rules; each having a particular shape. The shapes are, respectively C ⊢ D

• 1. F(C) ⊢ F(D)

C ⊢ F(D)

• 2. C ⊢ G(D)

F(C) ⊢ D

• 3. C ⊢ G(D)

When we read these as instructions to build arrows, we have that rules of type 1 are given by functoriality rules of type 2 are given by naturality rules of type 3 are given by adjointness (Note: we also have rules of type 2 with functors on the left)

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Framework

We have seen the general idea, let us try to be more precise. Definition A logic is a category U having categories as objects and functors as arrows; one of such object is called P, it is the category having formulas as objects and proofs as arrows. This seems a rather abstract and weak definition. However, this already ensure that, being P a category,

1

there is a proof A ⊢IdA A for each formula A, the identity proof

2

we can compose proofs if target and source match, the Cut A ⊢x B B ⊢y C A ⊢y◦x C

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Framework

We have seen the general idea, let us try to be more precise. Definition A logic is a category U having categories as objects and functors as arrows; one of such object is called P, it is the category having formulas as objects and proofs as arrows. This seems a rather abstract and weak definition. However, this already ensure that, being P a category,

1

there is a proof A ⊢IdA A for each formula A, the identity proof

2

we can compose proofs if target and source match, the Cut A ⊢x B B ⊢y C A ⊢y◦x C

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Framework

In this framework the features of a logic depend on the following additional assumptions: the objects in U and the closure under specific categorical constructions (e.g. the product) determine the “sorts”of the logic the arrows in U determines the connectives of the logic the relations between these arrows (seen as functors), give the rules of the logic

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Example

Minimal Logic

To see how this work we will look at an example: Minimal Logic. Assume U contains P and is closed under products, terminal objects and ()op, the functor that flips all the arrows in the category. Assume the existence of the functors: ∧ : P × P → P ∨ : Pop × Pop → Pop →: Pop × P → P ¬ : Pop → P ⊤ : 1 → P Note that we cannot just define these functors: we need to assume the existence of enough arrows in P in order to have functoriality.

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Example

Minimal Logic

To see how this work we will look at an example: Minimal Logic. Assume U contains P and is closed under products, terminal objects and ()op, the functor that flips all the arrows in the category. Assume the existence of the functors: ∧ : P × P → P ∨ : Pop × Pop → Pop →: Pop × P → P ¬ : Pop → P ⊤ : 1 → P Note that we cannot just define these functors: we need to assume the existence of enough arrows in P in order to have functoriality.

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Example

Minimal Logic

In order to have the usual rules for Minimal Logic, we need to buy something more, namely the mutual relations between such functors. We assume the adjunctions ∨ ⊣ ∆ ⊣ ∧ A∧ ⊣ A →, for all formulas A in P ⊤∧ ⊣ Id (Id is the identity functor) and the natural isomorphism (here displayed in the components A, B) (A → B) ∧ (A → ¬B) ≃ ¬A These give us the usual rules of minimal logic (examples at the blackboard).

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Example

Minimal Logic

In order to have the usual rules for Minimal Logic, we need to buy something more, namely the mutual relations between such functors. We assume the adjunctions ∨ ⊣ ∆ ⊣ ∧ A∧ ⊣ A →, for all formulas A in P ⊤∧ ⊣ Id (Id is the identity functor) and the natural isomorphism (here displayed in the components A, B) (A → B) ∧ (A → ¬B) ≃ ¬A These give us the usual rules of minimal logic (examples at the blackboard).

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Conclusions

Advantages

The framework we sketched has the following advantages: Modularity: Each connective is introduced in isolation, and we are extremely free in drawing the mutual relations between the connectives (but we are constrained by the categorical contructions available in U). Behaviourism: We do not define a language starting from atomic propositions, we merely define possible constructions on formulas (functors) and their effects on proofs (adjoints, nat. transformations). Cut Elimination: Being close to display calculi, it is easy in principle to check the premises of Belnap’s Theorem. Reduction: If the project is feasible, the framework allows for a completely categorical treatment of syntax and proof theory.

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Conclusions

Advantages

The framework we sketched has the following advantages: Modularity: Each connective is introduced in isolation, and we are extremely free in drawing the mutual relations between the connectives (but we are constrained by the categorical contructions available in U). Behaviourism: We do not define a language starting from atomic propositions, we merely define possible constructions on formulas (functors) and their effects on proofs (adjoints, nat. transformations). Cut Elimination: Being close to display calculi, it is easy in principle to check the premises of Belnap’s Theorem. Reduction: If the project is feasible, the framework allows for a completely categorical treatment of syntax and proof theory.

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Conclusions

Advantages

The framework we sketched has the following advantages: Modularity: Each connective is introduced in isolation, and we are extremely free in drawing the mutual relations between the connectives (but we are constrained by the categorical contructions available in U). Behaviourism: We do not define a language starting from atomic propositions, we merely define possible constructions on formulas (functors) and their effects on proofs (adjoints, nat. transformations). Cut Elimination: Being close to display calculi, it is easy in principle to check the premises of Belnap’s Theorem. Reduction: If the project is feasible, the framework allows for a completely categorical treatment of syntax and proof theory.

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Conclusions

Advantages

The framework we sketched has the following advantages: Modularity: Each connective is introduced in isolation, and we are extremely free in drawing the mutual relations between the connectives (but we are constrained by the categorical contructions available in U). Behaviourism: We do not define a language starting from atomic propositions, we merely define possible constructions on formulas (functors) and their effects on proofs (adjoints, nat. transformations). Cut Elimination: Being close to display calculi, it is easy in principle to check the premises of Belnap’s Theorem. Reduction: If the project is feasible, the framework allows for a completely categorical treatment of syntax and proof theory.

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Conclusions

Future work

Some ideas for what we will do next: Explore the capabilities of the framework in expressing more complex logics, such as modal logic or first order logic. Investigate how this work relates to Categorical Semantics and Functorial Model Theory. Study how the framework is connected with the other works in Categorical Proof Theory and the issue of identity of proofs. Can the framework give insights into Proof Theoretic Semantics?

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